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Algebra is hard

 

For certain values of k and m, the system

3a + 2b = 2

6a + 2b = k + 3a + mb

has infinitely many solutions (a,b).  What are k and m?

 May 7, 2023
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We can only have infinitely many solutions (a,b) when both of our equations are identical. We have:

3a + 2b = 2 

6a + 2b = k + 3a + mb 

If we subtract 3a from both sides of our lower equation we get 

3a + 2b = k + mb

 

Now, the left-hand side of both equations is identical.

 

For the right-hand side, we can set:

 k+mb=2

We can see that m must be 0, as if it is not we will have a variable with b, so our result will vary based on b, and not always be 2. This cannot be so we need m=0 as it will cancel out b. Then, we are left with k+0(b)=2    k=2.

 

Hence, we have that m=0 and k=2.

 

I hope this helped! 

 May 8, 2023
edited by Guest  May 8, 2023

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